> For the complete documentation index, see [llms.txt](https://dgtnetwork.gitbook.io/dgt-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://dgtnetwork.gitbook.io/dgt-docs/5.-exploring-tokenomics/5.7-token-price-simulation/5.7.1-nelson-siegel-svensson-model.md). # 5.7.1 Nelson-Siegel-Svensson model The **Nelson-Siegel-Svensson (NSS)** model (Medvedev 2019) is an extended version of the Nelson-Siegel (NS) model (Annaert et al. 2013), commonly used to describe interest rate term structures or yield curves. A yield curve represents the relationship between the interest rate (or cost of borrowing) and the time to maturity of the debt for a given borrower in each currency. The NSS model defines the instantaneous forward rate as a function of four components: a long-term factor, a short-term factor, a medium-term factor, and an over-demand factor: $$ Y(t)= β\_0 + \frac{β\_1 (1-e^{-t/τ\_1})}{t/τ\_1 } + β\_2\left( \frac{1-e^{-t/τ\_1}}{t/τ\_1} -e^{-t/τ\_1} \right) + β\_3 \left( \frac{1-e^{-t/τ\_2}}{t/τ\_2} - e^{-t/τ\_2} \right) = Y\_0+\\+Y\_1+Y\_2+Y\_3 \ \ \ \ \ \ \ \ \tag{40} $$ Here's what each component represents: o ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FdJBBaNvf3WHzRfCnx69H%2Fimage.png?alt=media\&token=fcafc244-c37f-42bb-a6ab-88ee371cb77c) (**Long term factor**): This component, represented by ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FfplucvEt1dyxNsEATUzf%2Fimage.png?alt=media\&token=98ce84f8-b417-4d2c-83c9-5d27c467ae15), is the long-term interest rate or the level factor of the yield curve. It is a constant value that sets the asymptotic future interest rate as time tends to infinity. o ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FDodPpzDppcoQSKjayeNx%2Fimage.png?alt=media\&token=895a2ca8-7f82-4ea5-be63-b489fa6417ef) (**Short term factor**): This component captures the impact of short-term rates and is responsible for the initial steepness of the yield curve. The ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FdxEM4kz7WU2BtBRhWcKu%2Fimage.png?alt=media\&token=cca95da4-f017-449b-9a15-bab714ed4b66) parameter sets the slope at time zero and ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2F0ehdKx2joPZYsSNEIMTi%2Fimage.png?alt=media\&token=cccb97b6-d996-4e57-b91d-330ab4988db7) represents a decay factor. The term ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FqKcZQq10sr49SANFIbys%2Fimage.png?alt=media\&token=eadc2af8-3612-482a-aae3-bfe0175ac7a9) introduces a decay effect, meaning the influence of this factor decreases as time progresses. o ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FPKcHRgYO9p6yg9jlFQ14%2Fimage.png?alt=media\&token=b2cae547-42ca-4e8a-a1df-3860f9518058) (**Medium term factor**): This component models the hump in the yield curve. ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2Fac85XL43F8b2JkKJiCIi%2Fimage.png?alt=media\&token=c3374e25-1034-4f1f-8f6b-ebced81a75da)sets the size of the hump, while ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FKovzU9XQ61o8swdoPS0w%2Fimage.png?alt=media\&token=d166b7c7-6e1c-423a-bb50-3279371eb089) again is a decay factor. The term ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FcHCtQyTIRd30Cld8EKzE%2Fimage.png?alt=media\&token=c5b927ae-b18f-43bc-8f5e-d47ff2564a89) means the influence of this factor increases initially, reaches a peak, and then decreases. o ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FNFGYvWJJFift4fhfXoKz%2Fimage.png?alt=media\&token=20028851-ba16-4ff4-8f14-07976d924960) (**Over-demand factor**): This component, unique to the NSS model, captures any additional influences on the interest rate not covered by the other three components. ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FLnREkKMt8Q8yjCfVKZel%2Fimage.png?alt=media\&token=e5b21c46-db12-4715-ae14-39229be04b81) sets the size of this influence, while ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FUl40mTj8EFFriMYDTRJj%2Fimage.png?alt=media\&token=ff221cac-3e2d-4cb5-8bc1-352c580164e3) acts as a decay factor. Like ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FPKcHRgYO9p6yg9jlFQ14%2Fimage.png?alt=media\&token=b2cae547-42ca-4e8a-a1df-3860f9518058), the term ![](https://3210145376-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FSvanFNAf6Jf8kwvqcdRQ%2Fuploads%2FdkJdnrKD0p6ilb2zv6zm%2Fimage.png?alt=media\&token=22d19cc0-4038-4804-a094-61211899c78f) introduces an increase, peak, and decrease effect. The final yield curve is the sum of these four components, yielding a flexible model that can capture a variety of yield curve shapes.

Figure 91 Adjusted NSS Model over time